Recall theorem 1 of [1]} gives for \(0\le k \le 2\)
\(\int _{T}^{2T} |\zeta (\tfrac{1}{2} + i t)|^{2k} dt \ll T(\log T)^{k^2}.\) 
Therefore by Hölder's inequality with \(p = \tfrac{1}{h}\)  and \(q = \tfrac{1}{1-h}\) , this estimate and Theorem REF  give
\(I_T (k,h)\le \Bigg (\int _T^{2T}|\zeta (\tfrac{1}{2} + i t)|^{2k - 2} |\zeta ^{\prime }(\tfrac{1}{2} + i t)|^2 &dt \Bigg ) ^{h} \left(\int _T^{2T} |\zeta (\tfrac{1}{2} + i t)|^{2k} dt\right)^{1-h}\\&\ll T(\log T)^{k^2 + 2 h}.\) 

The case of the joint moments of \(Z(t)\)  is similar since \(|Z(t)| = |\zeta (\tfrac{1}{2} + i t)|\) .
